Excuse me ... .

The definition of derivative a function respect to is

... .

Consider ... then

... .

In real calculus, we have known that ... .

So, we have to say that

... (*) ... .

The problem is how to prove the last equality (*) ... .

Maybe, we can prove the last equality by L'Hopital Theorem ... .

But, the L' Hopital Theorem uses the concept of derivative ... .

Whereas, the concept of derivative uses the concept of limit ... .

How can the last equality (*) be proven without L'Hopital Theorem ... ?

Is the last equation (*) an assumption ... ?

Thank you very much ... .